How can you get a plane with a emission node to be brighter in the center?
Material
You can use a Gradient node with Object mapping to make a spherical gradient based on the location of the object origin:

The mapping node is just used to scale the texture up a bit.
The multiply node controls the overall strength.
Note that scaling in object mode will stretch the texture accordingly, however applying the scale or scaling in edit mode will not.
AO
It's also possible that your reference image was created with a very heavy Ambient Occlusion effect.
To achieve this:

Enable AO in Properties &gt; Render layers &gt; Passes

Adjust it and multiply it with the combined pass in the compositor:



Combined with the gradient: